In triangle $ABC,\angle BAC = 60^o, AB = 3a$ and $AC = 4a, (a > 0)$. Let $M$ be point on the segment $AB$ such that $AM =\frac13 AB, N$ be point on the side $AC$ such that $AN =\frac12AC$. Let $I$ be midpoint of $MN$. Determine the length of $BI$. A. $\frac{a\sqrt2}{19}$ B. $\frac{2a}{\sqrt{19}}$ C. $\frac{19a\sqrt{19}}{2}$ D. $\frac{19a}{\sqrt2}$ E. $\frac{a\sqrt{19}}{2}$

Denote by $\lfloor x\rfloor$ the greatest positive integer less than or equal to $x$. Let $m\ge2$ be an integer, and let $s$ be a real number between $0$ and $1$. Define an infinite sequence of real numbers $a_1, a_2, a_3,\ldots$ by setting $a_1 = s$ and $ak = ma_{k-1}-(m-1)\lfloor a_{k-1}\rfloor$ for all $k\ge2$. For example, if $m = 3$ and $s = \tfrac58$, then we get $a_1 = \tfrac58$, $a_2 = \tfrac{15}8$, $a_3 = \tfrac{29}8$, $a_4 = \tfrac{39}8$, and so on. Call the sequence $a_1, a_2, a_3,\ldots$ $\textbf{orderly}$ if we can find rational numbers $b, c$ such that $\lfloor a_n\rfloor = \lfloor bn + c\rfloor$ for all $n\ge1$. With the example above where $m = 3$ and $s = \tfrac58$, we get an orderly sequence since $\lfloor a_n\rfloor = \left\lfloor\tfrac{3n}2-\tfrac32\right\rfloor$ for all $n$. Show that if $s$ is an irrational number and $m\ge2$ is any integer, then the sequence $a_1, a_2, a_3,\ldots$ is $\textbf{not}$ an orderly sequence.

In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$. Determine the length of $AD$. The answer must be given as a fraction with integer numerator and denominator.

For a positive integer $n$, let $R(n)$ be the sum of the remainders when $n$ is divided by $1,2, \cdots , n$. For example, $R(4) = 0 + 0 + 1 + 0 = 1,$ $R(7) = 0 + 1 + 1 + 3 + 2 + 1 + 0 = 8$. Find all positive integers such that $R(n) = n-1$.

Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.

If the function $f$ is defined by \[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is $\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.

Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes. (N. Belukhov)

For a given integer $n\geq 3$, determine the range of values for the expression \[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\] over real numbers $x_1,x_2,\ldots,x_n \geq 1$ satisfying $|x_k - x_{k+1}| \leq 1$ for all $1\leq k \leq n-1$. Do also determine when the extremal values are achieved. (Dan Schwarz)

The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$. You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$. (a) Find $g(2021)$. (b) Find $g(1/2021)$.

Bianca has a rectangle whose length and width are distinct primes less than 100. Let $P$ be the perimeter of her rectangle, and let $A$ be the area of her rectangle. What is the least possible value of $\frac{P^2}{A}$?

In an acute triangle $ABC$ let $AD$ and $BE$ be altitudes, and $AP$ and $BQ$ be bisectors. Let $I$ and $O$ be centers of incircle and circumcircle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear.

In $ABC$ triangle $I$ is incenter and incircle of $ABC$ tangents to $BC,AC,AB$ at $D,E,F$, respectively. If $AI$ intersects $DE$ and $DF$ at $P$ and $Q$, prove that the circumcenter of $DPQ$ triangle is the midpoint of $BC$.

$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum? [i]Proposed by Oleksiy Masalitin[/i]

Alice has $100$ balls and $10$ buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let $b_i$ be the number of balls in the $i$th bucket, for $1\le i \le 10$. Compute the expected value of $\Sigma_{i=1}^{10}b_i^2$

Prove that $\forall x_{1},x_{2},\ldots,x_{2011},y_{1},y_{2},\ldots,y_{2011}\in\mathbb{Z_{+}}$ product: \[(2x_{1}^{2}+3y_{1}^{2})(2x_{2}^{2}+3y_{2}^{2})\ldots(2x_{2011}^{2}+3y_{2011}^{2})\] is not a perfect square.